A085343 Number of primes between sigma(n) and phi(n).

0, 2, 1, 3, 1, 4, 1, 4, 3, 5, 1, 7, 1, 6, 5, 7, 1, 9, 1, 9, 6, 7, 1, 13, 3, 8, 5, 11, 1, 16, 1, 12, 7, 10, 6, 19, 1, 10, 7, 18, 1, 19, 1, 15, 12, 12, 1, 24, 3, 16, 9, 16, 1, 23, 8, 21, 11, 15, 1, 33, 1, 14, 16, 20, 8, 26, 1, 19, 10, 25, 1, 35, 1, 19, 18, 23, 7, 30, 1, 31, 14, 18, 1, 39, 10, 19

Offset

1,2

Comments

a(p) = 1 for prime p > 2. Since phi(p) = p - 1 and sigma(p) = p + 1, the largest prime q < p - 1 must be the prime previous to p, while p itself is the largest prime less than p + 1 for p > 2. - Michael De Vlieger, Jan 22 2020

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Formula

a(n) = Pi(sigma(n)) - Pi(phi(n)) = A000720(A000203(n)) - A000720(A000010(n)).

a(n) = A070803(n) - A070804(n). - Antti Karttunen, Jan 22 2020

Example

n=12: sigma(12)=28, phi(n)=4, Pi(28)-Pi(4)=9-2=7.

Mathematica

Array[Subtract @@ PrimePi@{DivisorSigma[1, #], EulerPhi@ #} &, 86] (* Michael De Vlieger, Jan 22 2020 *)

Prog

(PARI) a(n) = primepi(sigma(n)) - primepi(eulerphi(n)); \\ Michel Marcus, Aug 29 2019

Crossrefs

Cf. A000720, A000010, A000203, A070803, A070804, A085122, A085341-A085347.

Sequence in context: A097019 A262999 A324933 * A049077 A180184 A330752

Adjacent sequences: A085340 A085341 A085342 * A085344 A085345 A085346

Keyword

nonn

Author

Labos Elemer, Jul 10 2003

Status

approved